Eulerian–Lagrangian method for simulation of cloud cavitation

Abstract

We present a coupled Eulerian-Lagrangian method to simulate cloud cavitation in a compressible liquid. The method is designed to capture the strong, volumetric oscillations of each bubble and the bubble-scattered acoustics. The dynamics of the bubbly mixture is formulated using volume-averaged equations of motion. The continuous phase is discretized on an Eulerian grid and integrated using a high-order, finite-volume weighted essentially non-oscillatory (WENO) scheme, while the gas phase is modeled as spherical, Lagrangian point-bubbles at the sub-grid scale, each of whose radial evolution is tracked by solving the Keller-Miksis equation. The volume of bubbles is mapped onto the Eulerian grid as the void fraction by using a regularization (smearing) kernel. In the most general case, where the bubble distribution is arbitrary, three-dimensional Cartesian grids are used for spatial discretization. In order to reduce the computational cost for problems possessing translational or rotational homogeneities, we spatially average the governing equations along the direction of symmetry and discretize the continuous phase on two-dimensional or axi-symmetric grids, respectively. We specify a regularization kernel that maps the three-dimensional distribution of bubbles onto the field of an averaged two-dimensional or axi-symmetric void fraction. A closure is developed to model the pressure fluctuations at the sub-grid scale as synthetic noise. For the examples considered here, modeling the sub-grid pressure fluctuations as white noise agrees a priori with computed distributions from three-dimensional simulations, and suffices, a posteriori, to accurately reproduce the statistics of the bubble dynamics. The numerical method and its verification are described by considering test cases of the dynamics of a single bubble and cloud cavitaiton induced by ultrasound fields. (Kazuki Maeda) arXiv:1712.00670v2 [physics.flu-dyn] 5 Dec 2017 such methods are the most suitable for simulations at the scales of bubbles for a short period of time, typically single events of the collapse of bubbles [17,18,19,20].For more complex bubbly mixtures with relatively low void fractions of O(10 −2 ), methods that solve volume-or ensemble-averaged equations of motion have been used to compute the propagation of acoustic and shock waves [21,22,23,24,25,26,27,28,29,30] and the dynamics of bubble clouds [31,32,33,34,35,36,37]. In the classical averaging approaches, the bubbly-mixture is treated as a continuous media. The volume of dispersed bubbles is converted to a continuous void fraction field in a control volume that contains a sufficiently large number of bubbles. The length-scale of the control volume (averaging length-scale) is larger than the characteristic inter-bubble distance. The dynamics of the gas-phase are closed by considering the averaged change in the volumetric oscillations of bubbles in response to pressure fluctuations in the mixture. Bubbles are typically modeled as spherical cavities, of which dynamics are described by ordinary...

Lagrangian statistics of pressure fluctuation events in homogeneous isotropic turbulence

“…The reported results provide useful quantitative data to predict the frequency, intensity and duration of pressure fluctuations experienced by very small particles that are passively transported by a turbulent flow. They can be used, for example, to inform modern models of numerical cavitation [25][26][27] . The behavior at higher Reynolds numbers and in other turbulent flows should certainly be explored to gain further understanding.…”

Modeling and Analysis Thin Wall Film Formation with Integrated Discrete and Continuous Phase Simulation Approach

“…10, where the energy and temperature are averaged from each phase, as in Eq. 11and (12). The droplet is assumed to mix adiabatically and instantly when it impacts on the film, as calculated in Eq.…”

“…Anez et.al (Anez et al, 2019) focused on the unresolved interface and tested the model against realistic experimental data, where indicated that models that are computationally less demanding and still accurate enough are required in an industrial environment. Maeda et.al (Maeda et al, 2018) presented a coupled Eulerian-Lagrangian method to simulate cloud cavitation in a compressible liquid for capturing the strong oscillations of each bubble. All these studies show the efficiency and promising application of the integrated discrete and continuous phase simulation approach.…”

Numerical Simulation of the Bubble Cloud Dynamics in an Ultrasound Field

“…We simulate the dynamics of bubbly-mixture using an Eulerian-Lagrangian method [9]. We discretize the governing equations on an axi-symmetric grid and integrate it using a finite volume, fifth-order WENO scheme.…”

“…The simulation uses a coupled Eulerian-Lagrangian method for bubbly-liquid that can capture the dynamics of individual spherical bubbles and the fine structure of the pressure field. The numerical method and its validation have been described elsewhere [9]. We use the following two kinds of ultrasound wave: 1 cycle of a sinusoidal wave with a wavelength of 30 mm, and 10 cycles of a sinusoidal wave with a wavelength of 5 mm.…”